Hurwitz equivalence for Lefschetz fibrations and their multisections

TL;DR

This paper characterizes when two Lefschetz fibrations with multisections are isomorphic using monodromy factorizations and extends Hurwitz equivalence, providing new insights into their classification and construction.

Contribution

It introduces a criterion for isomorphism of Lefschetz fibrations with multisections based on monodromy factorizations and extends Hurwitz equivalence, also deriving new examples of non-decomposable fibrations.

Findings

Isomorphism classes characterized by monodromy factorizations

Extended Hurwitz equivalence for Lefschetz fibrations with multisections

Constructed new examples of fibrations not expressible as fiber sums

Abstract

In this article, we characterize isomorphism classes of Lefschetz fibrations with multisections via their monodromy factorizations. We prove that two Lefschetz fibrations with multisections are isomorphic if and only if their monodromy factorizations in the relevant mapping class groups are related to each other by a finite collection of modifications, which extend the well-known Hurwitz equivalence. This in particular characterizes isomorphism classes of Lefschetz pencils. We then show that, from simple relations in the mapping class groups, one can derive new (and old) examples of Lefschetz fibrations which cannot be written as fiber sums of blown-up pencils.